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Congruences modulo powers of 11 for some partition functions


Author: Liuquan Wang
Journal: Proc. Amer. Math. Soc. 146 (2018), 1515-1528
MSC (2010): Primary 05A17; Secondary 11F03, 11F33, 11P83
DOI: https://doi.org/10.1090/proc/13858
Published electronically: December 4, 2017
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Abstract: Let $ R_{0}(N)$ be the Riemann surface of the congruence subgroup $ \Gamma _{0}(N)$ of $ \mathrm {SL}_{2}(\mathbb{Z})$. Using some properties of the field of meromorphic functions on $ R_{0}(11)$, we confirm a conjecture of H.H. Chan and P.C. Toh [J. Number Theory 130 (2010), pp. 1898-1913] about the partition function $ p(n)$. Moreover, we prove three infinite families of congruences modulo arbitrary powers of 11 for other partition functions, including 11-regular partitions and 11-core partitions.


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  • [1] George E. Andrews, Michael D. Hirschhorn, and James A. Sellers, Arithmetic properties of partitions with even parts distinct, Ramanujan J. 23 (2010), no. 1-3, 169-181. MR 2739210, https://doi.org/10.1007/s11139-009-9158-0
  • [2] Tom M. Apostol, Modular functions and Dirichlet series in number theory, Graduate Texts in Mathematics, No. 41, Springer-Verlag, New York-Heidelberg, 1976. MR 0422157
  • [3] A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J. 8 (1967), 14-32. MR 0205958, https://doi.org/10.1017/S0017089500000045
  • [4] A. O. L. Atkin, Ramanujan congruences for $ p_{-k}(n)$, Canad. J. Math. 20 (1968), 67-78; corrigendum, ibid. 21 (1968), 256. MR 0233777
  • [5] Nayandeep Deka Baruah and Kallol Nath, Some results on 3-cores, Proc. Amer. Math. Soc. 142 (2014), no. 2, 441-448. MR 3133986, https://doi.org/10.1090/S0002-9939-2013-11784-3
  • [6] Hei-Chi Chan, Ramanujan's cubic continued fraction and Ramanujan type congruences for a certain partition function, Int. J. Number Theory 6 (2010), no. 4, 819-834. MR 2661283, https://doi.org/10.1142/S1793042110003241
  • [7] Hei-Chi Chan and Shaun Cooper, Congruences modulo powers of 2 for a certain partition function, Ramanujan J. 22 (2010), no. 1, 101-117. MR 2610610, https://doi.org/10.1007/s11139-009-9197-6
  • [8] Heng Huat Chan and Pee Choon Toh, New analogues of Ramanujan's partition identities, J. Number Theory 130 (2010), no. 9, 1898-1913. MR 2653203, https://doi.org/10.1016/j.jnt.2010.02.017
  • [9] F. G. Garvan, A simple proof of Watson's partition congruences for powers of $ 7$, J. Austral. Math. Soc. Ser. A 36 (1984), no. 3, 316-334. MR 733905
  • [10] Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and $ t$-cores, Invent. Math. 101 (1990), no. 1, 1-17. MR 1055707, https://doi.org/10.1007/BF01231493
  • [11] Frank G. Garvan, Some congruences for partitions that are $ p$-cores, Proc. London Math. Soc. (3) 66 (1993), no. 3, 449-478. MR 1207544, https://doi.org/10.1112/plms/s3-66.3.449
  • [12] Basil Gordon, Ramanujan congruences for $ p_{-k}\ ({\rm mod}\,11^{r})$, Glasgow Math. J. 24 (1983), no. 2, 107-123. MR 706138, https://doi.org/10.1017/S0017089500005164
  • [13] B. Gordon and K. Hughes, Ramanujan congruences for $ q(n)$, Analytic number theory (Philadelphia, Pa., 1980) Lecture Notes in Math., vol. 899, Springer, Berlin-New York, 1981, pp. 333-359. MR 654539
  • [14] Basil Gordon and Ken Ono, Divisibility of certain partition functions by powers of primes, Ramanujan J. 1 (1997), no. 1, 25-34. MR 1607526, https://doi.org/10.1023/A:1009711020492
  • [15] Michael D. Hirschhorn and David C. Hunt, A simple proof of the Ramanujan conjecture for powers of $ 5$, J. Reine Angew. Math. 326 (1981), 1-17. MR 622342
  • [16] Michael D. Hirschhorn and James A. Sellers, Elementary proofs of various facts about 3-cores, Bull. Aust. Math. Soc. 79 (2009), no. 3, 507-512. MR 2505355, https://doi.org/10.1017/S0004972709000136
  • [17] Marvin I. Knopp, Modular functions in analytic number theory, Markham Publishing Co., Chicago, Ill., 1970. MR 0265287
  • [18] Louis W. Kolitsch, A congruence for generalized Frobenius partitions with $ 3$ colors modulo powers of $ 3$, Analytic number theory (Allerton Park, IL, 1989) Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 343-348. MR 1084189
  • [19] Jeremy Lovejoy and David Penniston, $ 3$-regular partitions and a modular $ K3$ surface, $ q$-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000) Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 177-182. MR 1874530, https://doi.org/10.1090/conm/291/04901
  • [20] Jeremy Lovejoy, Divisibility and distribution of partitions into distinct parts, Adv. Math. 158 (2001), no. 2, 253-263. MR 1822684, https://doi.org/10.1006/aima.2000.1967
  • [21] Jeremy Lovejoy, The number of partitions into distinct parts modulo powers of 5, Bull. London Math. Soc. 35 (2003), no. 1, 41-46. MR 1934430, https://doi.org/10.1112/S0024609302001492
  • [22] Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $ q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
  • [23] Peter Paule and Cristian-Silviu Radu, The Andrews-Sellers family of partition congruences, Adv. Math. 230 (2012), no. 3, 819-838. MR 2921161, https://doi.org/10.1016/j.aim.2012.02.026
  • [24] S. Ramanujan, Some properties of $ p(n)$, the number of partitions of $ n$ [Proc. Cambridge Philos. Soc. 19 (1919), 207-210], Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, pp. 210-213. MR 2280868
  • [25] Jacob Sturm, On the congruence of modular forms, Number theory (New York, 1984-1985) Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 275-280. MR 894516, https://doi.org/10.1007/BFb0072985
  • [26] Liuquan Wang, Arithmetic identities and congruences for partition triples with 3-cores, Int. J. Number Theory 12 (2016), no. 4, 995-1010. MR 3484295, https://doi.org/10.1142/S1793042116500627
  • [27] Liuquan Wang, Explicit formulas for partition pairs and triples with 3-cores, J. Math. Anal. Appl. 434 (2016), no. 2, 1053-1064. MR 3415707, https://doi.org/10.1016/j.jmaa.2015.09.074
  • [28] L. Wang, Congruences for 5-regular partitions modulo powers of 5, Ramanujan J., 44 (2017), no. 2, 343-358, doi:10.1007/s11139-015-9767-8.
  • [29] Liuquan Wang, Congruences modulo powers of 5 for two restricted bipartitions, Ramanujan J. 44 (2017), no. 3, 471-491. MR 3723436

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Additional Information

Liuquan Wang
Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, People’s Republic of China — and — Department of Mathematics, National University of Singapore, Singapore 119076, Singapore
Email: mathlqwang@163.com; wangliuquan@u.nus.edu

DOI: https://doi.org/10.1090/proc/13858
Keywords: Partitions, congruences modulo powers of 11, regular partitions, $t$-core partitions
Received by editor(s): October 29, 2016
Received by editor(s) in revised form: June 19, 2017
Published electronically: December 4, 2017
Dedicated: Dedicated to Professor Heng Huat Chan on the occasion of his 50th birthday
Communicated by: Ken Ono
Article copyright: © Copyright 2017 American Mathematical Society

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